
In the above \(S\)  and \(S^{\prime }\)  sum over the sublattices \(A\)  and \(B\) , and \(j\)  sums over the layers top and bottom. We also consider the possibility that the hopping constant depends not only on the displacement \(y + U^{\perp }_j(x) - U^{\perp }_{j^{\prime }}(x)\) , but also on the orientation of the nearest neighbor bonds [1]}. Without the lattice distortion, the hoppings are given by \(t^{j j^{\prime }}_{sym}\left[ y + U^\perp _{j}\left(x\right) - U^\perp _{j^{\prime }}(x),\delta _S, \delta _{S^{\prime }} \right]\) . In the presence of the lattice distortion, the hoppings can be expanded to the first order of the change of bond angles \(\delta \theta _{j, S}^{(\alpha )}\)  and \(\delta \theta _{j^{\prime }, S^{\prime }}^{(\alpha ^{\prime })}\)  where \(\alpha \)  and \(\alpha ^{\prime }\)  are the index of nearest neighbor bonds, ranging from 1 to 3. Thus, the correction is \( t^{(1)}_{j j^{\prime }, S}(y) \frac{1}{3} \sum _{\alpha = 1}^3 \delta \theta _{j,S}^{(\alpha )} + t^{(2)}_{j j^{\prime }, S^{\prime }}(y) \frac{1}{3}\sum _{\alpha ^{\prime } = 1}^3 \delta \theta _{j^{\prime }, S^{\prime }}^{(\alpha ^{\prime })}\) , where \(t^{(1)}_{j\ne j^{\prime },S}(y) = \frac{\partial t^{j\ne j^{\prime }}_{sym}}{\partial \theta _{j, S}}\)  and \(t^{(2)}_{j\ne j^{\prime },S^{\prime }}(y) = \frac{\partial t^{j\ne j^{\prime }}_{sym}}{\partial \theta _{j^{\prime },S^{\prime }}}\) .
